> Was Specker's result indeed the first of this kind?
Kleene predated this with his computable unbounded tree whose every
path is non-computable:
S.C. Kleene. Recursive functions and intuitionistic mathematics.
In L.M. Graves, E. Hille, P.A. Smith, and O. Zariski, editors,
Proceedings of the International Congress of Mathemaiticans,
August 1950. Cambridge, Mass., pages 679–685, 1952
With kind regards,
Andrej